### Heteroskedasticity

```Applied Econometrics
Applied Econometrics
Second edition
Dimitrios Asteriou and Stephen G. Hall
Applied Econometrics
HETEROSKEDASTICITY
1. What is Heteroskedasticity
2. Consequences of
Heteroskedasticity
3. Detecting Heteroskedasticity
4. Resolving Heteroskedasticity
Applied Econometrics
Learning Objectives
1. Understand the meaning of heteroskedasticity and
homoskedasticity through examples.
2. Understand the consequences of heteroskedasticity on OLS
estimates.
3. Detect heteroskedasticity through graph inspection.
4. Detect heteroskedasticity through formal econometric tests.
5. Distinguish among the wide range of available tests for
detecting heteroskedasticity.
6. Perform heteroskedasticity tests using econometric software.
7. Resolve heteroskedasticity using econometric software.
Applied Econometrics
What is Heteroskedasticity
Hetero (different or unequal) is the opposite of
Homo (same or equal)…
Applied Econometrics
What is Heteroskedasticity
Assumption 5 of the CLRM states that the
disturbances should have a constant (equal)
variance independent of t:
Var(ut)=σ2
Therefore, having an equal variance means
that the disturbances are homoskedastic.
Applied Econometrics
What is Heteroskedasticity
If the homoskedasticity assumption is
violated then
Var(ut)=σt2
Where the only difference is the subscript t,
attached to the σt2, which means that the
variance can change for every different
observation in the sample t=1, 2, 3, 4, …, n.
Look at the following graphs…
Applied Econometrics
What is Heteroskedasticity
Applied Econometrics
What is Heteroskedasticity
Applied Econometrics
What is Heteroskedasticity
Applied Econometrics
What is Heteroskedasticity
First graph: Homoskedastic residuals
Second graph: income-consumption
patterns, for low levels of income not much
choices, opposite for high levels.
Third graph: improvements in data
collection techniques (large banks) or to
error learning models (experience decreases
the chance of making large errors).
Applied Econometrics
Consequences of
Heteroskedasticity
1. The OLS estimators are still unbiased and consistent. This is
because none of the explanatory variables is correlated with the
error term. So a correctly specified equation will give us values
of estimated coefficient which are very close to the real
parameters.
2. Affects the distribution of the estimated coefficients increasing
the variances of the distributions and therefore making the OLS
estimators inefficient.
3. Underestimates the variances of the estimators, leading to higher
values of t and F statistics.
Applied Econometrics
Detecting Heteroskedasticity
There are two ways in general.
The first is the informal way which is done through graphs
and therefore we call it the graphical method.
The second is through formal tests for heteroskedasticity,
like the following ones:
1.
2.
3.
4.
5.
6.
The Breusch-Pagan LM Test
The Glesjer LM Test
The Harvey-Godfrey LM Test
The Park LM Test
The Goldfeld-Quandt Tets
White’s Test
Applied Econometrics
Detecting Heteroskedasticity
We plot the square of the obtained residuals against fitted Y
and the X’s and we see the patterns.
Applied Econometrics
Detecting Heteroskedasticity
Applied Econometrics
Detecting Heteroskedasticity
Applied Econometrics
Detecting Heteroskedasticity
Applied Econometrics
Detecting Heteroskedasticity
Applied Econometrics
The Breusch-Pagan LM Test
Step 1: Estimate the model by OLS and obtain the residuals
Step 2: Run the following auxiliary regression:
2
uˆ t  a 1  a 2 Z 2 t  a 3 Z 3 t  ...  a p Z pt  v t
Step 3: Compute LM=nR2, where n and R2 are from the
auxiliary regression.
Step 4: If LM-stat>χ2p-1 critical reject the null and conclude
that there is significant evidence of heteroskedasticity
Applied Econometrics
The Glesjer LM Test
Step 1: Estimate the model by OLS and obtain the residuals
Step 2: Run the following auxiliary regression:
| uˆ t | a 1  a 2 Z 2 t  a 3 Z 3 t  ...  a p Z pt  v t
Step 3: Compute LM=nR2, where n and R2 are from the
auxiliary regression.
Step 4: If LM-stat>χ2p-1 critical reject the null and conclude
that there is significant evidence of heteroskedasticity
Applied Econometrics
The Harvey-Godfrey LM Test
Step 1: Estimate the model by OLS and obtain the residuals
Step 2: Run the following auxiliary regression:
2
ˆ
ln u t  a 1  a 2 Z 2 t  a 3 Z 3 t  ...  a p Z pt  v t
Step 3: Compute LM=nR2, where n and R2 are from the
auxiliary regression.
Step 4: If LM-stat>χ2p-1 critical reject the null and conclude
that there is significant evidence of heteroskedasticity
Applied Econometrics
The Park LM Test
Step 1: Estimate the model by OLS and obtain the residuals
Step 2: Run the following auxiliary regression:
2
ln uˆ t  a 1  a 2 ln Z 2 t  a 3 ln Z 3 t  ...  a p ln Z pt  v t
Step 3: Compute LM=nR2, where n and R2 are from the
auxiliary regression.
Step 4: If LM-stat>χ2p-1 critical reject the null and conclude
that there is significant evidence of heteroskedasticity
Applied Econometrics
The Engle’s ARCH Test
Engle introduced a new concept allowing for heteroskedasticity to occur in the variance of the error terms, rather
than in the error terms themselves.
The key idea is that the variance of ut depends on the size of
the squarred error term lagged one period u2t-1 for the first
order model or:
Var(ut)=γ1+γ2u2t-1
The model can be easily extended for higher orders:
Var(ut)=γ1+γ2u2t-1+…+ γpu2t-p
Applied Econometrics
The Engle’s ARCH Test
Step 1: Estimate the model by OLS and obtain the residuals
Step 2: Regress the squared residuals to a constant and lagged
terms of squared residuals, the number of lags will be
determined by the hypothesized order of ARCH effects.
Step 3: Compute the LM statistic = (n-ρ)R2 from the LM model
and compare it with the chi-square critical value.
Step 4: Conclude
Applied Econometrics
The Goldfeld-Quandt Test
Step 1: Identify one variable that is closely related to
the variance of the disturbances, and order (rank) the
observations of this variable in descending order
(starting with the highest and going to the lowest).
Step 2: Split the ordered sample into two equally sized
sub-samples by omitting c central observations, so
that the two samples will contain ½(n-c)
observations.
Applied Econometrics
The Goldfeld-Quandt Test
Step 3:Run and OLS regression of Y on the X
variable that you have used in step 1 for each subsample and obtain the RSS for each equation.
is the RSS with the largest value.
Step 5: If F-stat>F-crit(1/2(n-c)-l,1/2(n-c)-k) reject the null
of homoskedasticity.
Applied Econometrics
The White’s Test
Step 1: Estimate the model by OLS and obtain the residuals
Step 2: Run the following auxiliary regression:
2
2
2
uˆ t  a1  a 2 X 2 t  a 3 X 3 t  a 4 X 2 t  a 5 X 3 t  a 6 X 2 t X 3 t  v t
Step 3: Compute LM=nR2, where n and R2 are from the
auxiliary regression.
Step 4: If LM-stat>χ2p-1 critical reject the null and conclude
that there is significant evidence of heteroskedasticity
Applied Econometrics
Resolving Heteroskedasticity
We have three different cases:
(a) Generalized Least Squares
(b) Weighted Least Squares
(c) Heteroskedasticity-Consistent Estimation
Methods
Applied Econometrics
Generalized Least Squares
Consider the model
Yt=β1+β2X2t+β3X3t+β4X4t+…+βkXkt+ut
where
Var(ut)=σt2
Applied Econometrics
Generalized Least Squares
If we divide each term by the standard deviation of the
error term, σt we get:
Yt=β1 (1/σt) +β2X2t/σt +β3X3t/σt +…+βkXkt/σt +ut/σt
or
Y*t= β*1+ β*2X*2t+ β*3X*3t+…+ β*kX*kt+u*t
Where we have now that:
Var(u*t)=Var(ut/σt)=Var(ut)/σt2=1
Applied Econometrics
Weighted Least Squares
The GLS procedure is the same as the WLS where we
have weights, wt, adjusting our variables.
Define wt=1/σt, and rewrite the original model as:
wtYt=β1wt+β2X2twt+β3X3twt+…+βkXktwt+utwt
Where if we define as wtYt-1=Y*t and Xitwt=X*it
we get
Y*t= β*1+ β*2X*2t+ β*3X*3t+…+ β*kX*kt+u*t
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